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The question asks you to graph the function \( h(x) = (x + 2)^2 + 5 \) by using transformations of the standard quadratic function \( f(x) = x^2 \).

### Step-by-Step Solution:

1. **Start with the base function**: \( f(x) = x^2 \), which is a parabola opening upwards with its vertex at the origin (0, 0).

2. **Apply the horizontal shift**: The expression \( (x + 2) \) indicates a horizontal shift. Since it is \( x + 2 \), the graph shifts 2 units to the left.

3. **Apply the vertical shift**: The \( +5 \) outside of the squared term indicates a vertical shift upwards by 5 units.

### Determining the Correct Graph:

- The vertex of the new function \( h(x) = (x + 2)^2 + 5 \) will be at \( (-2, 5) \).
- The graph should be a parabola that opens upwards, shifted left 2 units and up 5 units from the original \( x^2 \) graph.

### Looking at the Options:

- **Option A**: Shows a graph with the vertex at (-2, 5), which matches the transformation.
- **Options B, C, and D**: Do not match the correct shifts of left 2 units and up 5 units.

### Conclusion:

The correct answer is **A**.

Would you like a deeper explanation of transformations or have any other questions? Here are some related questions that could help you understand transformations better:

1. What are the different types of transformations for graphs?
2. How do you determine the direction of shifts from a function's equation?
3. What is the effect of multiplying a function by a negative number?
4. How does a vertical stretch or compression affect the graph of a function?
5. What is the difference between a horizontal shift and a vertical shift?

**Tip:** Remember that a positive number inside the parentheses shifts the graph to the left, and a positive number outside shifts it upwards.

Begin by graphing the standard quadratic function f(x) = x^2. Then use transformations of this graph to graph the given function h(x) = (x + 2)^2 + 5.

Learn how to graph the function h(x) = (x + 2)^2 + 5 by applying transformations to the standard quadratic function f(x) = x^2. This problem involves understanding horizontal and vertical shifts in quadratic graphs. Suitable for high school students in Grades 9-12.

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